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1: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
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4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

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See also:

Annotations for §4.31 and Ch.4

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4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

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See also:

Annotations for §4.31 and Ch.4

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4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

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See also:

Annotations for §4.31 and Ch.4

2: 4.4 Special Values and Limits

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§4.4(iii) Limits

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4.4.13 $$\underset{x\to \mathrm{\infty }}{lim}{x}^{-a}\ln x=0,$$ $\mathrm{\Re }a>0$,

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Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.1.30

Permalink:

http://dlmf.nist.gov/4.4.E13

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Annotations for §4.4(iii), §4.4 and Ch.4

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4.4.14 $$\underset{x\to 0}{lim}{x}^a\ln x=0,$$ $\mathrm{\Re }a>0$,

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Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.1.31

Permalink:

http://dlmf.nist.gov/4.4.E14

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Annotations for §4.4(iii), §4.4 and Ch.4

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4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

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Annotations for §4.4(iii), §4.4 and Ch.4

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3: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

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Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

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Annotations for §4.17 and Ch.4

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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

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See also:

Annotations for §4.17 and Ch.4

4: 10.30 Limiting Forms

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§10.30(i) $z\to 0$

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5: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

6: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

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See also:

Annotations for §29.5 and Ch.29

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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

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Annotations for §29.5 and Ch.29

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7: 18.11 Relations to Other Functions

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Jacobi

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18.11.5 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(1-\frac{z^2}{2n^2}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(\cos \frac{z}{n}\right)=\frac{2^{\alpha }}{z^{\alpha }}{J}_{\alpha }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.1

Referenced by:

§18.11(ii), §18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E5

Encodings:

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See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

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Laguerre

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Hermite

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18.11.8 $$\underset{n\to \mathrm{\infty }}{lim}\frac{{(-1)}^n}{2^{2n}n!}{H}_{2n+1}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{2}{{\pi }^{\frac{1}{2}}}\sin z.$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.4

Referenced by:

§18.11(ii), §18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E8

Encodings:

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Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

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8: 10.52 Limiting Forms

§10.52 Limiting Forms

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9: 10.7 Limiting Forms

§10.7 Limiting Forms

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10: 28.12 Definitions and Basic Properties

… ►However, these functions are not the limiting values of ${\mathrm{me}}_{\pm \nu }(z,q)$ as $\nu \to n$ $(\ne 0)$. … ►Again, the limiting values of ${\mathrm{ce}}_{\nu }(z,q)$ and ${\mathrm{se}}_{\nu }(z,q)$ as $\nu \to n$ $(\ne 0)$ are not the functions ${\mathrm{ce}}_n(z,q)$ and ${\mathrm{se}}_n(z,q)$ defined in §28.2(vi). …